Question: $f(x, y) = \cos(xy) + y^3x^2$ What is $\nabla \cdot (\nabla f)$ ? $\nabla \cdot (\nabla f) = $
Solution: The Laplacian of a scalar field $f$ is the sum of each of its second partial derivatives. $\nabla \cdot (\nabla f) = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2}$ [What does it mean to take a dot product with the gradient?] Let's find the second partial derivatives of $f$ ! $\begin{aligned} f_{xx} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right] \\ \\ &= \dfrac{\partial}{\partial x} \left[ -y\sin(xy) + 2y^3x \right] \\ \\ &= -y^2\cos(xy) + 2y^3\\ \\ f_{yy} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial y} \right] \\ \\ &= \dfrac{\partial}{\partial y} \left[ -x\sin(xy) + 3y^2x^2 \right] \\ \\ &= -x^2\cos(xy) + 6yx^2 \end{aligned}$ The Laplacian is $\nabla \cdot (\nabla f) = f_{xx} + f_{yy}$. Therefore: $\nabla \cdot (\nabla f) = -\cos(xy)(x^2 + y^2) + 2y^3 + 6yx^2$